From French Patent Application 91-00567 there is already known a device for performing an attachment of a seat cover to a vehicle seat, comprising a plurality of preassembled pieces of elastic, each having a hook at a first end, and at a second end, a strip of relatively rigid and thin plastic material adapted to be sewn onto the covering, this seam being the only operation to be performed whereby the attachment is positioned on the covering. The latter is installed on the seat by being disposed upon or over the seat, and then engaging the hooks on a rigid element of the seat, generally one of the springs situated on the underside of the base of the seat.